Laplace transforms worksheet with answers

Free printable Laplace transforms worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (s)f(t)t = 1f(1) = ?

Find the inverse Laplace transform f(t) of F(s) = 12s/(s² + 36) with a table of transform pairs, and evaluate f(1).

F(s) = 12s/(s² + 36)t = 1 s
f(1) =
2.Easy
t (s)f(t)

A signal f(t) has the Laplace transform F(s) = (−5s + 1)/(s² + 2s + 27). Find the initial slope f′(0⁺) with the initial value theorem.

F(s) = (−5s + 1)/(s² + 2s + 27)
f′(0⁺) =
3.Medium
ReIm−1−2−3−4−5−61

F(s) = (2s + 10)/(s² + 8s + 12). Its inverse transform has the form f(t) = A·e−2t + B·e−6t. Find the coefficient A by partial fractions.

F(s) = (2s + 10)/(s² + 8s + 12)
A =
4.Medium
t (s)f(t)t = 0.25f(0.25) = ?

Find f(t) for F(s) = (4s + 1)/(s² + 4s + 4) and evaluate f(0.25).

F(s) = (4s + 1)/(s² + 4s + 4)t = 0.25 s
f(0.25) =
5.Hard
t (s)y(t)t = 0.25y(0.25) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 6 and y′(0) = 5 using the Laplace transform, and find y(0.25).

y″ + 7y′ + 12y = 0y(0) = 6y′(0) = 5t = 0.25 s
y(0.25) =
6.Hard
ReIm5

A system is described by the differential equation y″ + 6y′ + 34y = 4u′ + 15u, with input u(t) and output y(t). Find its transfer function G(s) = Y(s)/U(s) (zero initial conditions) and evaluate the impulse response g(t) at t = 0.25 s. Find g(0.25).

y″ + 6y′ + 34y = 4u′ + 15ut = 0.25 s
g(0.25) =

Answer key

  1. 1. f(1) = 11.52
  2. 2. f′(0⁺) = 11
  3. 3. A = 1.5
  4. 4. f(0.25) = 1.365
  5. 5. y(0.25) = 5.237
  6. 6. g(0.25) = 0.8648

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